Violation of the incompressibility of liquid by simple shear flow

Abstract

In standard fluid dynamics1,2, the density change associated with flow is often assumed to be negligible, implying that the fluid is incompressible. For example, this has been established for simple shear flows, where no pressure change is associated with flow: there is no volume deformation due to viscous stress and inertial effects can be neglected. Accordingly, any flow-induced instabilities (such as cavitation) are unexpected for simple shear flows. Here we demonstrate that the incompressibility condition can be violated even for simple shear flows, by taking into account the coupling between the flow and density fluctuations, which arises owing to the density dependence of the viscosity. We show that a liquid can become mechanically unstable above a critical shear rate that is given by the inverse of the derivative of viscosity with respect to pressure. Our model predicts that, for very viscous liquids, this shear-induced instability should occur at moderate shear rates that are experimentally accessible. Our results explain the unusual shear-induced instability observed in viscous lubricants3,4,5, and may illuminate other poorly understood phenomena associated with mechanical instability of liquids at low Reynolds number; for example, shear-induced cavitation and bubble growth6, and shear-banding of very viscous liquids such as metallic glasses7 and the Earth's mantle8.

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Figure 1: Intuitive explanation for the mechanism of shear-induced instability of a liquid.
Figure 2: Shear-induced instability of a thermodynamically stable liquid.
Figure 3: Rheological behaviour of a thermodynamically stable liquid under shear deformation.

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Acknowledgements

We are grateful to C. P. Royall for a critical reading of our manuscript. This work was partially supported by a grant-in-aid for JSPS Fellows (A.F.) and a grant-in-aid from the Ministry of Education, Culture, Sports, Science and Technology, Japan (H.T.).

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Correspondence to Akira Furukawa or Hajime Tanaka.

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Furukawa, A., Tanaka, H. Violation of the incompressibility of liquid by simple shear flow. Nature 443, 434–438 (2006). https://doi.org/10.1038/nature05119

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